New Quantum Error-Correcting Codes from Hermitian Self-Orthogonal Codes over GF(4)

  • Jon-Lark Kim


In order to construct good quantum-error-correcting codes, we construct good Hermitian self-orthogonal linear codes over GF(4). In this paper we construct record-breaking pure quantum-error-correcting codes of length 24 with 2 encoded qubits and minimum weight 7 from Hermitian self-orthogonal codes of length 24 with dimension 11 over GF(4). This shows that length n = 24 is the smallest length for any known [[n, k, d]] quantum-error-correcting code with k ≥ 2 and d = 7. We also give a construction method to produce Hermitian self-orthogonal linear codes GF(4) from a shorter length such code.


Linear Code Minimum Weight Cyclic Code Association Scheme Quantum Code 
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  1. 1.
    Calderbank, A. R., Rains, E. M., Shor, P. W., Sloane, N. J. A. (1998) Quantum error correction via codes over GF(4). IEEE Trans. Inform. Theory. 44, 1369–1387MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Gaborit, P., Huffman, W. C., Kim, J.-L., and Pless, V. (2001) On additive GF(4) codes. DIMACS Workshop on Codes and Association Schemes DIMACS Series in Discrete Math. and Theoret. Computer Science, American Mathematical Society, 56, 135–149MathSciNetGoogle Scholar
  3. 3.
    Kim, J.-L. (2001) New self-dual codes over GF(4) with the highest known minimum weights. IEEE Trans. Inform. Theory. 47, 1575–1580MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Shor, P. W. (1995) Scheme for reducing decoherence in quantum memory. Phys. Rev. A. 52, 2493CrossRefGoogle Scholar
  5. 5.
    Thangaraj, A., McLaughlin. S. W. (2001) Quantum codes from cyclic codes over GF(4m). IEEE Trans. Inform. Theory. 47. 1176–1178MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jon-Lark Kim
    • 1
  1. 1.Department of Mathematics, Statistics, and Computer Science, 322 SEO(M/C 249)University of Illinois-ChicagoChicagoUSA

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